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Symmetry
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Conservation Laws and Symmetries of Differential Equations
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Conservation Laws and Symmetries of Differential Equations

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 January 2020) | Viewed by 22714

Special Issue Editors

Prof. Dr. Maria Luz Gandarias

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Guest Editor
Department of Mathematics, Cádiz University, 11003 Cádiz, Spain
Interests: symmetry methods; applications for differential equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Maria Santos Bruzón Gallego

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Co-Guest Editor
Department of Mathematics, University of Cádiz, 11510 Cádiz, Spain
Interests: group analysis; methods of group transformation: classical symmetries; nonclassical methods; direct methods and conservation laws applied to ordinary differential equations; partial differential equations and systems of partial differential equations
Special Issues, Collections and Topics in MDPI journals
Dr. Rita Tracinà

E-Mail Website
Co-Guest Editor
Department of Mathematics and Computer Science, Catania University, 95125 Catania, Italy
Interests: symmetry methods; applications for differential equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Mariano Torrisi

E-Mail Website
Co-Guest Editor
Dipartimento di Matematica e Informatica, University of Catania, Catania, Italy
Interests: group methods for nonlinear differential equations (both ODEs and PDEs); reduction techniques for the search of exact solutions of PDEs; applications of the group methods to reaction diffusion models, such as nonlinear governing equations modeling population dynamics and biomathematical problems; nonlinear diffusion and propagation of heat
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Prof. Dr. Chaudry Masood Khalique

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Co-Guest Editor
International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa
Interests: nonlinear differential equations; Lie symmetry method; closed-form solutions; conservation laws; mathematical physics; analytical solution methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Conservation laws play a vital role in the reduction and solution process of the differential equations.  It is well known that the integrability of the differential equations is strongly related to the existence of conservation laws. Conservation laws are used for existence, uniqueness and stability analysis and for the development of numerical methods. Recently, they have been applied to find exact solutions of certain partial differential equations.

Symmetry analysis for differential equations was developed by Sophus Lie in the latter half of the nineteenth century.  It systematically unifies and extends the well-known ad hoc methods to construct closed form solutions for differential equations, in particular for nonlinear differential equations. These methods are highly algorithmic and hence responsive to symbolic computation.

Conservation laws and symmetry analysis have applications to genuine physical systems of differential equations that are found in diverse fields as continuum mechanics, classical mechanics, quantum mechanics, relativity, numerical analysis, tumour growth, finance, and economics and so on.

The main aim of this Special Issue is to focus on some recent developments in methods and applications of conservation laws and symmetries of differential equations. Mathematicians, engineers, physicists and other scientists for whom differential equations are treasured research apparatuses are encouraged to submit their research to this special issue.

Prof. Maria Luz Gandarias
Prof. Chaudry Masood Khalique
Prof. Mariano Torrisi
Assist. Prof. Rita Tracinà
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Differential equations
  • Conservation laws
  • Exact solutions
  • Symmetry

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Published Papers (8 papers)

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Research

15 pages, 298 KiB  
Article
Lie Symmetries and Low-Order Conservation Laws of a Family of Zakharov-Kuznetsov Equations in 2 + 1 Dimensions
by María S. Bruzón, Tamara M. Garrido, Elena Recio and Rafael de la Rosa
Symmetry 2020, 12(8), 1277; https://doi.org/10.3390/sym12081277 - 2 Aug 2020
Cited by 2 | Viewed by 2183
Abstract
In this work, we study a generalised (2+1) equation of the Zakharov–Kuznetsov (ZK)(m,n,k) equation involving three arbitrary functions. From the point of view of the Lie symmetry theory, we have derived all [...] Read more.
In this work, we study a generalised (2+1) equation of the Zakharov–Kuznetsov (ZK)(m,n,k) equation involving three arbitrary functions. From the point of view of the Lie symmetry theory, we have derived all Lie symmetries of this equation depending on the arbitrary functions. Line soliton solutions have also been obtained. Moreover, we study the low-order conservation laws by applying the multiplier method. This family of equations is rich in Lie symmetries and conservation laws. Finally, when the equation is expressed in potential form, it admits a variational structure in the case when two of the arbitrary functions are linear. In addition, the corresponding Hamiltonian formulation is presented. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
11 pages, 251 KiB  
Article
Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions
by María Luz Gandarias, María Rosa Durán and Chaudry Masood Khalique
Symmetry 2020, 12(6), 950; https://doi.org/10.3390/sym12060950 - 4 Jun 2020
Cited by 19 | Viewed by 2277
Abstract
In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for [...] Read more.
In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
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13 pages, 241 KiB  
Article
Conditional Lie–Bäcklund Symmetries and Functionally Generalized Separation of Variables to Quasi-Linear Diffusion Equations with Source
by Rui Wang and Lina Ji
Symmetry 2020, 12(5), 844; https://doi.org/10.3390/sym12050844 - 21 May 2020
Cited by 2 | Viewed by 2126
Abstract
The conditional Lie–Bäcklund symmetry method is applied to investigate the functionally generalized separation of variables for quasi-linear diffusion equations with a source. The equations and the admitted conditional Lie–Bäcklund symmetries related to invariant subspaces are identified. The exact solutions possessing the form of [...] Read more.
The conditional Lie–Bäcklund symmetry method is applied to investigate the functionally generalized separation of variables for quasi-linear diffusion equations with a source. The equations and the admitted conditional Lie–Bäcklund symmetries related to invariant subspaces are identified. The exact solutions possessing the form of the functionally generalized separation of variables are constructed for the resulting equations due to the corresponding symmetry reductions. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
24 pages, 8024 KiB  
Article
A New Approach in Analytical Dynamics of Mechanical Systems
by Iuliu Negrean, Adina-Veronica Crișan and Sorin Vlase
Symmetry 2020, 12(1), 95; https://doi.org/10.3390/sym12010095 - 3 Jan 2020
Cited by 34 | Viewed by 3773
Abstract
This paper presents a new approach to the advanced dynamics of mechanical systems. It is known that in the movements corresponding to some mechanical systems (e.g., robots), accelerations of higher order are developed. Higher-order accelerations are an integral part of higher-order acceleration energies. [...] Read more.
This paper presents a new approach to the advanced dynamics of mechanical systems. It is known that in the movements corresponding to some mechanical systems (e.g., robots), accelerations of higher order are developed. Higher-order accelerations are an integral part of higher-order acceleration energies. Unlike other research papers devoted to these advanced notions, the main purpose of the paper is to present, in a matrix form, the defining expressions for the acceleration energies of a higher order. Following the differential principle in generalized form (a generalization of the Lagrange–D’Alembert principle), the equations of the dynamics of fast-moving systems include, instead of kinetic energies, the acceleration energies of higher-order. To establish the equations which characterize both the energies of accelerations and the advanced dynamics, the following input parameters are considered: matrix exponentials and higher-order differential matrices. An application of a 5 d.o.f robot structure is presented in the final part of the paper. This is used to illustrate the validity of the presented mathematical formulations. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
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9 pages, 264 KiB  
Article
First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach
by Muhammad Umar Farooq, Chaudry Masood Khalique and Fazal M. Mahomed
Symmetry 2019, 11(10), 1244; https://doi.org/10.3390/sym11101244 - 4 Oct 2019
Cited by 1 | Viewed by 2081
Abstract
The aim of the present work is to classify the Noether-like operators of two-dimensional physical systems whose dynamics is governed by a pair of Lane-Emden equations. Considering first-order Lagrangians for these systems, we construct corresponding first integrals. It is seen that for a [...] Read more.
The aim of the present work is to classify the Noether-like operators of two-dimensional physical systems whose dynamics is governed by a pair of Lane-Emden equations. Considering first-order Lagrangians for these systems, we construct corresponding first integrals. It is seen that for a number of forms of arbitrary functions appearing in the set of equations, the Noether-like operators also fulfill the classical Noether symmetry condition for the pairs of real Lagrangians and the generated first integrals are reminiscent of those we obtain from the complex Lagrangian approach. We also investigate the cases in which the underlying systems are reducible via quadrature. We derive some interesting results about the nonlinear systems under consideration and also find that the algebra of Noether-like operators is Abelian in a few cases. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
18 pages, 12970 KiB  
Article
Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method
by Imtiaz Ahmad, Muhammad Ahsan, Iltaf Hussain, Poom Kumam and Wiyada Kumam
Symmetry 2019, 11(3), 394; https://doi.org/10.3390/sym11030394 - 18 Mar 2019
Cited by 27 | Viewed by 3726
Abstract
In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial [...] Read more.
In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial discretization, which convert the mentioned partial differential equations into a system of ordinary differential equations. The obtained system has been solved by the forward Euler difference formula. An upwind technique is utilized in the case of the convection-dominated coupled Burgers’ model equation. Having no need for the mesh in the problem domain and being less sensitive to the variation of the shape parameter as compared to global meshless methods are the salient features of the local meshless method. Both rectangular and non-rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy, and the ease of implementation of the proposed method are shown via test problems. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
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9 pages, 5988 KiB  
Article
Nonlocal Symmetries for Time-Dependent Order Differential Equations
by Andrei Ludu
Symmetry 2018, 10(12), 771; https://doi.org/10.3390/sym10120771 - 19 Dec 2018
Cited by 1 | Viewed by 2917
Abstract
A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order [...] Read more.
A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
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25 pages, 380 KiB  
Article
New Stability Criterion for the Dissipative Linear System and Analysis of Bresse System
by Yoshihiro Ueda
Symmetry 2018, 10(11), 542; https://doi.org/10.3390/sym10110542 - 25 Oct 2018
Cited by 8 | Viewed by 2643
Abstract
In this article, we introduce a new approach to obtain the property of the dissipative structure for a system of differential equations. If the system has a viscosity or relaxation term which possesses symmetric property, Shizuta and Kawashima in 1985 introduced the suitable [...] Read more.
In this article, we introduce a new approach to obtain the property of the dissipative structure for a system of differential equations. If the system has a viscosity or relaxation term which possesses symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition called in this article Classical Stability Condition for the corresponding eigenvalue problem of the system, and derived the detailed relation between the coefficient matrices of the system and the eigenvalues. However, there are some complicated physical models which possess a non-symmetric viscosity or relaxation term and we cannot apply Classical Stability Condition to these models. Under this situation, our purpose in this article is to extend Classical Stability Condition for complicated models and to make the relation between the coefficient matrices and the corresponding eigenvalues clear. Furthermore, we shall explain the new dissipative structure through the several concrete examples. Full article
(This article belongs to the Special Issue Conservation Laws and Symmetries of Differential Equations)
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